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Theorem List for Intuitionistic Logic Explorer - 501-600   *Has distinct variable group(s)
TypeLabelDescription
Statement
 
Theoremad2ant2lr 501 Deduction adding two conjuncts to antecedent. (Contributed by NM, 23-Nov-2007.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( th  /\  ph )  /\  ( ps 
 /\  ta ) )  ->  ch )
 
Theoremad2ant2rl 502 Deduction adding two conjuncts to antecedent. (Contributed by NM, 24-Nov-2007.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ph  /\ 
 th )  /\  ( ta  /\  ps ) ) 
 ->  ch )
 
Theoremadantl3r 503 Deduction adding 1 conjunct to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  ->  ta )   =>    |-  ( ( ( ( ( ph  /\  et )  /\  ps )  /\  ch )  /\  th )  ->  ta )
 
Theoremad4ant13 504 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( (
 ph  /\  th )  /\  ps )  /\  ta )  ->  ch )
 
Theoremad4ant14 505 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( (
 ph  /\  th )  /\  ta )  /\  ps )  ->  ch )
 
Theoremad4ant23 506 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( th  /\  ph )  /\  ps )  /\  ta )  ->  ch )
 
Theoremad4ant24 507 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( th  /\  ph )  /\  ta )  /\  ps )  ->  ch )
 
Theoremadantl4r 508 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  ( ( ( ( ( ph  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ka )   =>    |-  ( ( ( ( ( ( ph  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ka )
 
Theoremad5ant12 509 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  ps )  /\  th )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant13 510 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ps )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant14 511 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ta )  /\  ps )  /\  et )  ->  ch )
 
Theoremad5ant15 512 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( ph  /\  th )  /\  ta )  /\  et )  /\  ps )  ->  ch )
 
Theoremad5ant23 513 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ps )  /\  ta )  /\  et )  ->  ch )
 
Theoremad5ant24 514 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ta )  /\  ps )  /\  et )  ->  ch )
 
Theoremad5ant25 515 Deduction adding conjuncts to antecedent. (Contributed by Alan Sare, 17-Oct-2017.) (Proof shortened by Wolf Lammen, 14-Apr-2022.)
 |-  ( ( ph  /\  ps )  ->  ch )   =>    |-  ( ( ( ( ( th  /\  ph )  /\  ta )  /\  et )  /\  ps )  ->  ch )
 
Theoremadantl5r 516 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  ( ( ( ( ( ( ph  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ka )   =>    |-  (
 ( ( ( ( ( ( ph  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ka )
 
Theoremadantl6r 517 Deduction adding 1 conjunct to antecedent. (Contributed by Thierry Arnoux, 11-Feb-2018.)
 |-  ( ( ( ( ( ( ( ph  /\ 
 et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ka )   =>    |-  (
 ( ( ( ( ( ( ( ph  /\ 
 ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ka )
 
Theoremsimpll 518 Simplification of a conjunction. (Contributed by NM, 18-Mar-2007.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  ph )
 
Theoremsimplr 519 Simplification of a conjunction. (Contributed by NM, 20-Mar-2007.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  ps )
 
Theoremsimprl 520 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  ps )
 
Theoremsimprr 521 Simplification of a conjunction. (Contributed by NM, 21-Mar-2007.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  ch )
 
Theoremsimplll 522 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  ->  ph )
 
Theoremsimpllr 523 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  ->  ps )
 
Theoremsimplrl 524 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ( ph  /\  ( ps  /\  ch ) )  /\  th )  ->  ps )
 
Theoremsimplrr 525 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ( ph  /\  ( ps  /\  ch ) )  /\  th )  ->  ch )
 
Theoremsimprll 526 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ph  /\  (
 ( ps  /\  ch )  /\  th ) ) 
 ->  ps )
 
Theoremsimprlr 527 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ph  /\  (
 ( ps  /\  ch )  /\  th ) ) 
 ->  ch )
 
Theoremsimprrl 528 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  ->  ch )
 
Theoremsimprrr 529 Simplification of a conjunction. (Contributed by Jeff Hankins, 28-Jul-2009.)
 |-  ( ( ph  /\  ( ps  /\  ( ch  /\  th ) ) )  ->  th )
 
Theoremsimp-4l 530 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  -> 
 ph )
 
Theoremsimp-4r 531 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  ->  ps )
 
Theoremsimp-5l 532 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  ->  ph )
 
Theoremsimp-5r 533 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  ->  ps )
 
Theoremsimp-6l 534 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  ->  ph )
 
Theoremsimp-6r 535 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  ->  ps )
 
Theoremsimp-7l 536 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  ->  ph )
 
Theoremsimp-7r 537 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  ->  ps )
 
Theoremsimp-8l 538 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  ->  ph )
 
Theoremsimp-8r 539 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  ->  ps )
 
Theoremsimp-9l 540 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  -> 
 ph )
 
Theoremsimp-9r 541 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ( ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  ->  ps )
 
Theoremsimp-10l 542 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  -> 
 ph )
 
Theoremsimp-10r 543 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ( ( ph  /\ 
 ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  ->  ps )
 
Theoremsimp-11l 544 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  /\  ka )  -> 
 ph )
 
Theoremsimp-11r 545 Simplification of a conjunction. (Contributed by Mario Carneiro, 4-Jan-2017.)
 |-  ( ( ( ( ( ( ( ( ( ( ( (
 ph  /\  ps )  /\  ch )  /\  th )  /\  ta )  /\  et )  /\  ze )  /\  si )  /\  rh )  /\  mu )  /\  la )  /\  ka )  ->  ps )
 
Theorempm4.87 546 Theorem *4.87 of [WhiteheadRussell] p. 122. (Contributed by NM, 3-Jan-2005.) (Proof shortened by Eric Schmidt, 26-Oct-2006.)
 |-  ( ( ( ( ( ph  /\  ps )  ->  ch )  <->  ( ph  ->  ( ps  ->  ch )
 ) )  /\  (
 ( ph  ->  ( ps 
 ->  ch ) )  <->  ( ps  ->  (
 ph  ->  ch ) ) ) )  /\  ( ( ps  ->  ( ph  ->  ch ) )  <->  ( ( ps 
 /\  ph )  ->  ch )
 ) )
 
Theorema2and 547 Deduction distributing a conjunction as embedded antecedent. (Contributed by AV, 25-Oct-2019.) (Proof shortened by Wolf Lammen, 19-Jan-2020.)
 |-  ( ph  ->  (
 ( ps  /\  rh )  ->  ( ta  ->  th ) ) )   &    |-  ( ph  ->  ( ( ps 
 /\  rh )  ->  ch )
 )   =>    |-  ( ph  ->  (
 ( ( ps  /\  ch )  ->  ta )  ->  ( ( ps  /\  rh )  ->  th )
 ) )
 
Theoremanimpimp2impd 548 Deduction deriving nested implications from conjunctions. (Contributed by AV, 21-Aug-2022.)
 |-  ( ( ps  /\  ph )  ->  ( ch  ->  ( th  ->  et )
 ) )   &    |-  ( ( ps 
 /\  ( ph  /\  th ) )  ->  ( et 
 ->  ta ) )   =>    |-  ( ph  ->  ( ( ps  ->  ch )  ->  ( ps  ->  ( th  ->  ta ) ) ) )
 
Theoremabai 549 Introduce one conjunct as an antecedent to the other. "abai" stands for "and, biconditional, and, implication". (Contributed by NM, 12-Aug-1993.) (Proof shortened by Wolf Lammen, 7-Dec-2012.)
 |-  ( ( ph  /\  ps ) 
 <->  ( ph  /\  ( ph  ->  ps ) ) )
 
Theoreman12 550 Swap two conjuncts. Note that the first digit (1) in the label refers to the outer conjunct position, and the next digit (2) to the inner conjunct position. (Contributed by NM, 12-Mar-1995.)
 |-  ( ( ph  /\  ( ps  /\  ch ) )  <-> 
 ( ps  /\  ( ph  /\  ch ) ) )
 
Theoreman32 551 A rearrangement of conjuncts. (Contributed by NM, 12-Mar-1995.) (Proof shortened by Wolf Lammen, 25-Dec-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch ) 
 <->  ( ( ph  /\  ch )  /\  ps ) )
 
Theoreman13 552 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
 |-  ( ( ph  /\  ( ps  /\  ch ) )  <-> 
 ( ch  /\  ( ps  /\  ph ) ) )
 
Theoreman31 553 A rearrangement of conjuncts. (Contributed by NM, 24-Jun-2012.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch ) 
 <->  ( ( ch  /\  ps )  /\  ph )
 )
 
Theoreman12s 554 Swap two conjuncts in antecedent. The label suffix "s" means that an12 550 is combined with syl 14 (or a variant). (Contributed by NM, 13-Mar-1996.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ps  /\  ( ph  /\  ch )
 )  ->  th )
 
Theoremancom2s 555 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ph  /\  ( ch  /\  ps ) ) 
 ->  th )
 
Theoreman13s 556 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ch  /\  ( ps  /\  ph )
 )  ->  th )
 
Theoreman32s 557 Swap two conjuncts in antecedent. (Contributed by NM, 13-Mar-1996.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ps )  ->  th )
 
Theoremancom1s 558 Inference commuting a nested conjunction in antecedent. (Contributed by NM, 24-May-2006.) (Proof shortened by Wolf Lammen, 24-Nov-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ps 
 /\  ph )  /\  ch )  ->  th )
 
Theoreman31s 559 Swap two conjuncts in antecedent. (Contributed by NM, 31-May-2006.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  th )   =>    |-  ( ( ( ch 
 /\  ps )  /\  ph )  ->  th )
 
Theoremanass1rs 560 Commutative-associative law for conjunction in an antecedent. (Contributed by Jeff Madsen, 19-Jun-2011.)
 |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  th )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ps )  ->  th )
 
Theoremanabs1 561 Absorption into embedded conjunct. (Contributed by NM, 4-Sep-1995.) (Proof shortened by Wolf Lammen, 16-Nov-2013.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ph )  <->  (
 ph  /\  ps )
 )
 
Theoremanabs5 562 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 9-Dec-2012.)
 |-  ( ( ph  /\  ( ph  /\  ps ) )  <-> 
 ( ph  /\  ps )
 )
 
Theoremanabs7 563 Absorption into embedded conjunct. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 17-Nov-2013.)
 |-  ( ( ps  /\  ( ph  /\  ps )
 ) 
 <->  ( ph  /\  ps ) )
 
Theoremanabsan 564 Absorption of antecedent with conjunction. (Contributed by NM, 24-Mar-1996.) (Revised by NM, 18-Nov-2013.)
 |-  ( ( ( ph  /\  ph )  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabss1 565 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 31-Dec-2012.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ph )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabss4 566 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.)
 |-  ( ( ( ps 
 /\  ph )  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabss5 567 Absorption of antecedent into conjunction. (Contributed by NM, 10-May-1994.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
 |-  ( ( ph  /\  ( ph  /\  ps ) ) 
 ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabsi5 568 Absorption of antecedent into conjunction. (Contributed by NM, 11-Jun-1995.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
 |-  ( ph  ->  (
 ( ph  /\  ps )  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremanabsi6 569 Absorption of antecedent into conjunction. (Contributed by NM, 14-Aug-2000.)
 |-  ( ph  ->  (
 ( ps  /\  ph )  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremanabsi7 570 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 18-Nov-2013.)
 |-  ( ps  ->  (
 ( ph  /\  ps )  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremanabsi8 571 Absorption of antecedent into conjunction. (Contributed by NM, 26-Sep-1999.)
 |-  ( ps  ->  (
 ( ps  /\  ph )  ->  ch ) )   =>    |-  ( ( ph  /\ 
 ps )  ->  ch )
 
Theoremanabss7 572 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 19-Nov-2013.)
 |-  ( ( ps  /\  ( ph  /\  ps )
 )  ->  ch )   =>    |-  (
 ( ph  /\  ps )  ->  ch )
 
Theoremanabsan2 573 Absorption of antecedent with conjunction. (Contributed by NM, 10-May-2004.) (Revised by NM, 1-Jan-2013.)
 |-  ( ( ph  /\  ( ps  /\  ps ) ) 
 ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoremanabss3 574 Absorption of antecedent into conjunction. (Contributed by NM, 20-Jul-1996.) (Proof shortened by Wolf Lammen, 1-Jan-2013.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ps )  ->  ch )   =>    |-  ( ( ph  /\  ps )  ->  ch )
 
Theoreman4 575 Rearrangement of 4 conjuncts. (Contributed by NM, 10-Jul-1994.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) )  <-> 
 ( ( ph  /\  ch )  /\  ( ps  /\  th ) ) )
 
Theoreman42 576 Rearrangement of 4 conjuncts. (Contributed by NM, 7-Feb-1996.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) )  <-> 
 ( ( ph  /\  ch )  /\  ( th  /\  ps ) ) )
 
Theoreman4s 577 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ( ps  /\  th ) ) 
 ->  ta )
 
Theoreman42s 578 Inference rearranging 4 conjuncts in antecedent. (Contributed by NM, 10-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ch  /\  th ) ) 
 ->  ta )   =>    |-  ( ( ( ph  /\ 
 ch )  /\  ( th  /\  ps ) ) 
 ->  ta )
 
Theoremanandi 579 Distribution of conjunction over conjunction. (Contributed by NM, 14-Aug-1995.)
 |-  ( ( ph  /\  ( ps  /\  ch ) )  <-> 
 ( ( ph  /\  ps )  /\  ( ph  /\  ch ) ) )
 
Theoremanandir 580 Distribution of conjunction over conjunction. (Contributed by NM, 24-Aug-1995.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ch ) 
 <->  ( ( ph  /\  ch )  /\  ( ps  /\  ch ) ) )
 
Theoremanandis 581 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
 |-  ( ( ( ph  /\ 
 ps )  /\  ( ph  /\  ch ) ) 
 ->  ta )   =>    |-  ( ( ph  /\  ( ps  /\  ch ) ) 
 ->  ta )
 
Theoremanandirs 582 Inference that undistributes conjunction in the antecedent. (Contributed by NM, 7-Jun-2004.)
 |-  ( ( ( ph  /\ 
 ch )  /\  ( ps  /\  ch ) ) 
 ->  ta )   =>    |-  ( ( ( ph  /\ 
 ps )  /\  ch )  ->  ta )
 
Theoremsyl2an2 583 syl2an 287 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ch  /\  ph )  ->  th )   &    |-  ( ( ps 
 /\  th )  ->  ta )   =>    |-  (
 ( ch  /\  ph )  ->  ta )
 
Theoremsyl2an2r 584 syl2anr 288 with antecedents in standard conjunction form. (Contributed by Alan Sare, 27-Aug-2016.)
 |-  ( ph  ->  ps )   &    |-  (
 ( ph  /\  ch )  ->  th )   &    |-  ( ( ps 
 /\  th )  ->  ta )   =>    |-  (
 ( ph  /\  ch )  ->  ta )
 
Theoremimpbida 585 Deduce an equivalence from two implications. (Contributed by NM, 17-Feb-2007.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 ch )  ->  ps )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
 
Theorempm3.45 586 Theorem *3.45 (Fact) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  ->  ps )  ->  ( ( ph  /\  ch )  ->  ( ps  /\  ch )
 ) )
 
Theoremim2anan9 587 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  ( ( ph  /\  th )  ->  ( ( ps 
 /\  ta )  ->  ( ch  /\  et ) ) )
 
Theoremim2anan9r 588 Deduction joining nested implications to form implication of conjunctions. (Contributed by NM, 29-Feb-1996.)
 |-  ( ph  ->  ( ps  ->  ch ) )   &    |-  ( th  ->  ( ta  ->  et ) )   =>    |-  ( ( th  /\  ph )  ->  ( ( ps  /\  ta )  ->  ( ch  /\  et )
 ) )
 
Theoremanim12dan 589 Conjoin antecedents and consequents in a deduction. (Contributed by Mario Carneiro, 12-May-2014.)
 |-  ( ( ph  /\  ps )  ->  ch )   &    |-  ( ( ph  /\ 
 th )  ->  ta )   =>    |-  (
 ( ph  /\  ( ps 
 /\  th ) )  ->  ( ch  /\  ta )
 )
 
Theorempm5.1 590 Two propositions are equivalent if they are both true. Theorem *5.1 of [WhiteheadRussell] p. 123. (Contributed by NM, 21-May-1994.)
 |-  ( ( ph  /\  ps )  ->  ( ph  <->  ps ) )
 
Theorempm3.43 591 Theorem *3.43 (Comp) of [WhiteheadRussell] p. 113. (Contributed by NM, 3-Jan-2005.) (Revised by NM, 27-Nov-2013.)
 |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  ->  ( ph  ->  ( ps  /\ 
 ch ) ) )
 
Theoremjcab 592 Distributive law for implication over conjunction. Compare Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Apr-1994.) (Proof shortened by Wolf Lammen, 27-Nov-2013.)
 |-  ( ( ph  ->  ( ps  /\  ch )
 ) 
 <->  ( ( ph  ->  ps )  /\  ( ph  ->  ch ) ) )
 
Theorempm4.76 593 Theorem *4.76 of [WhiteheadRussell] p. 121. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  ->  ps )  /\  ( ph  ->  ch ) )  <->  ( ph  ->  ( ps  /\  ch )
 ) )
 
Theorempm4.38 594 Theorem *4.38 of [WhiteheadRussell] p. 118. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ( ph  <->  ch )  /\  ( ps  <->  th ) )  ->  ( ( ph  /\  ps ) 
 <->  ( ch  /\  th ) ) )
 
Theorembi2anan9 595 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 31-Jul-1995.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ta  <->  et ) )   =>    |-  ( ( ph  /\ 
 th )  ->  (
 ( ps  /\  ta ) 
 <->  ( ch  /\  et ) ) )
 
Theorembi2anan9r 596 Deduction joining two equivalences to form equivalence of conjunctions. (Contributed by NM, 19-Feb-1996.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ta  <->  et ) )   =>    |-  ( ( th  /\  ph )  ->  ( ( ps  /\  ta )  <->  ( ch  /\  et )
 ) )
 
Theorembi2bian9 597 Deduction joining two biconditionals with different antecedents. (Contributed by NM, 12-May-2004.)
 |-  ( ph  ->  ( ps 
 <->  ch ) )   &    |-  ( th  ->  ( ta  <->  et ) )   =>    |-  ( ( ph  /\ 
 th )  ->  (
 ( ps  <->  ta )  <->  ( ch  <->  et ) ) )
 
Theorempm5.33 598 Theorem *5.33 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  /\  ( ps  ->  ch ) )  <->  ( ph  /\  (
 ( ph  /\  ps )  ->  ch ) ) )
 
Theorempm5.36 599 Theorem *5.36 of [WhiteheadRussell] p. 125. (Contributed by NM, 3-Jan-2005.)
 |-  ( ( ph  /\  ( ph 
 <->  ps ) )  <->  ( ps  /\  ( ph  <->  ps ) ) )
 
Theorembianabs 600 Absorb a hypothesis into the second member of a biconditional. (Contributed by FL, 15-Feb-2007.)
 |-  ( ph  ->  ( ps 
 <->  ( ph  /\  ch ) ) )   =>    |-  ( ph  ->  ( ps  <->  ch ) )
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